Show that $T: \Bbb R^n \to \Bbb R^m$ defined by $T(u)=Mu$ (multiplication by $M$, an $m \times n$ matrix) is a linear combination.
I know I need to prove that the function preserves vector addition and scalar multiplication by showing that:
1) $T(v_1 + v_2) = T(v_1) + T(v_2)$
2) $T(\lambda v) = \lambda T(v)$
but I don't know how to start doing this. Any help would be greatly appreciated!
Figured it out myself: a lot simpler than I was thinking.
$T(u_1 + u_2) = M(u_1+u_2) = Mu_1 + Mu_2 = T(u_1) + T(u_2)$
Therefore, vector addition is preserved.
$T(\lambda u) = M(\lambda u) = \lambda (Mu) = \lambda T(u)$
Therefore, scalar multiplication is preserved and T is a linear transformation.